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Multiplying and Dividing Rational Expressions !Attendance Problems. Simplify each expression. Assume all variables are nonzero.

1. ! 2. ! 3. ! 4. !

!!Factor each expression.

5. ! 6. ! 7. ! !!!!!• I can simplify rational expressions. • I can multiply and divide rational expressions. !Vocabulary: Rational Expression !Common Core: CCSS.MATH.CONTENT.HSA.APR.D.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. !You worked with inverse variation functions such as ! . The expression on the

right side of this equation is a rational expression. A rational expression is a quotient of two polynomials. Other examples of rational expressions include the following:

!

x5 ⋅ x2 y3 ⋅ y3 x6

x2y2

y5

x2 − 2x − 8 x2 − 5x x5 − 9x3

y = 5x

Algebra II 5-2 Study Guide Page ! of !2 14

Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator.

!

When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.

Caution!

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Video Example 1. Simplify. Identify any x-values for which the expression is undefined.

A. ! B. !

!!!!!!!!!!!

3t 4

15tx2 − 25

x2 − 3x −10You have worked with inverse variation functions such as y = 5 __ x . The expression on the right side of this equation is a rational expression. A rational expression is a quotient of two polynomials. Other examples of rational expressions include the following:

x 2 - 4 _ x + 2

10 _ x 2 - 6

x + 3 _ x - 7

Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator.

1E X A M P L E Simplifying Rational Expressions

Simplify. Identify any x-values for which the expression is undefined.

A 3 x 7 _ 2 x 4

3 _ 2

x 7 - 4 = 3 _ 2

x 3 Quotient of Powers Property

The expression is undefined at x = 0 because this value of x makes 2 x 4 equal 0.

B x 2 - 2x - 3 __ x 2 + 5x + 4

(x - 3) (x + 1)

__ (x + 1) (x + 4)

= (x - 3)

_ (x + 4)

Factor; then divide out common factors.

The expression is undefined at x = -1 and x = -4 because these values of x make the factors (x + 1) and (x + 4) equal 0.

Check Substitute x = -1 and x = -4 into the original expression.

(-1) 2 - 2 (-1) - 3

__ (-1) 2 + 5 (-1) + 4

= 0

_ 0

(-4) 2 - 2 (-4) - 3

__ (-4) 2 + 5 (-4) + 4

= 21

_ 0

Both values of x result in division by 0, which is undefined.

Simplify. Identify any x-values for which the expression is undefined.

1a. 16 x 11 _ 8 x 2

1b. 3x + 4 __ 3 x 2 + x - 4

1c. 6 x 2 + 7x + 2 __ 6 x 2 - 5x - 6

Multiplying and Dividing Rational Expressions

ObjectivesSimplify rational expressions.

Multiply and divide rational expressions.

Vocabularyrational expression

Why learn this?You can simplify rational expressions to determine the probability of hitting an archery target. (See Exercise 35.)

9 _ 24

= 3 · 3 _ 8 · 3

= 3 _ 8

When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.

Des

ign

Pic

s In

c/A

lam

y D

esig

n P

ics

Inc/

Ala

my

5-2 Multiplying and Dividing Rational Expressions 321

5-2CC.9-12.A.APR.7 Understand that rational expressions form a system analogous to the rational numbers … add, subtract, multiply, and divide rational expressions.

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Example 1. Simplify. Identify any x-values for which the expression is undefined.

A. ! B. !

!!!!!!!!!!!Guided Practice. Simplify. Identify any x-values for which the expression is undefined.

8. ! 9. ! 10. !

!!!!!!!!!!!!!

10x8

6x4x2 + x − 2x2 + 2x − 3

16x11

8x23x + 4

3x2 + x − 46x2 + 7x + 26x2 − 5x − 6

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Video Example 2. Simplify ! . Identify any x-values for which the

expression is undefined. !!!!!!!!!!!

5x − x2

x2 − x − 20

2E X A M P L E Simplifying by Factoring -1

Simplify 2x - x 2 _______ x 2 - x - 2

. Identify any x-values for which the expression is undefined.

-1 ( x 2 - 2x) __

x 2 - x - 2

Factor out –1 in the numerator so that x 2 is positive, and reorder the terms.

-1 (x) (x - 2) __

(x - 2) (x + 1)

Factor the numerator and denominator. Divide out common factors.

-x _ x + 1

Simplify.

The expression is undefined at x = 2 and x = -1.

Check The calculator screens suggest that 2x - x 2 ________

x 2 - x - 2 = -x ____ x + 1 except

when x = 2 or x = -1.

Simplify. Identify any x-values for which the expression is undefined.

2a. 10 - 2x _ x - 5

2b. - x 2 + 3x __ 2 x 2 - 7x + 3

You can multiply rational expressions the same way that you multiply fractions.

Multiplying Rational Expressions

1. Factor all numerators and denominators completely.

2. Divide out common factors of the numerators and denominators.

3. Multiply numerators. Then multiply denominators.

4. Be sure the numerator and denominator have no common factors other than 1.

3E X A M P L E Multiplying Rational Expressions

Multiply. Assume that all expressions are defined.

A 2 x 4 y 5

_ 3 x 2

· 15 x 2 _ 8 x 3 y 2

B x + 2 _ 3x + 12

· x + 4 _ x 2 - 4

2 x 4 1 y 5 3

_ 3 x 2

· 515 x 2 _

48 x 3 y 2 x + 2 _

3 (x + 4) · x + 4 __

(x + 2) (x - 2)

5x y 3

_ 4

1 _ 3 (x - 2)

or 1 _ 3x - 6

Multiply. Assume that all expressions are defined.

3a. x _ 15

· x 7 _ 2x

· 20 _ x 4

3b. 10x - 40 __ x 2 - 6x + 8

· x + 3 _ 5x + 15

2

322 Chapter 5 Rational and Radical Functions

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Example 2. Simplify ! . Identify any x-values for which the expression is

undefined. !!!!!!!!!!!!!Guided Practice. Simplify. Identify any x-values for which the expression is undefined.

11. ! 12. !

!!!!!!!!!!!!You can multiply rational expressions the same way that you multiply fractions. !!

4x − x2

x2 − 2x − 8

10 − 2xx − 5

−x2 + 3x2x2 − 7x + 3

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Video Example 3. Multiply. Assume that all expressions are defined.

A. ! B. !

!!!!!!!!!

!

3x2w3

4w2 ⋅ 20x6

21xw4x + 35x + 30

⋅ x + 6x2 − 9

2E X A M P L E Simplifying by Factoring -1

Simplify 2x - x 2 _______ x 2 - x - 2

. Identify any x-values for which the expression is undefined.

-1 ( x 2 - 2x) __

x 2 - x - 2

Factor out –1 in the numerator so that x 2 is positive, and reorder the terms.

-1 (x) (x - 2) __

(x - 2) (x + 1)

Factor the numerator and denominator. Divide out common factors.

-x _ x + 1

Simplify.

The expression is undefined at x = 2 and x = -1.

Check The calculator screens suggest that 2x - x 2 ________

x 2 - x - 2 = -x ____ x + 1 except

when x = 2 or x = -1.

Simplify. Identify any x-values for which the expression is undefined.

2a. 10 - 2x _ x - 5

2b. - x 2 + 3x __ 2 x 2 - 7x + 3

You can multiply rational expressions the same way that you multiply fractions.

Multiplying Rational Expressions

1. Factor all numerators and denominators completely.

2. Divide out common factors of the numerators and denominators.

3. Multiply numerators. Then multiply denominators.

4. Be sure the numerator and denominator have no common factors other than 1.

3E X A M P L E Multiplying Rational Expressions

Multiply. Assume that all expressions are defined.

A 2 x 4 y 5

_ 3 x 2

· 15 x 2 _ 8 x 3 y 2

B x + 2 _ 3x + 12

· x + 4 _ x 2 - 4

2 x 4 1 y 5 3

_ 3 x 2

· 515 x 2 _

48 x 3 y 2 x + 2 _

3 (x + 4) · x + 4 __

(x + 2) (x - 2)

5x y 3

_ 4

1 _ 3 (x - 2)

or 1 _ 3x - 6

Multiply. Assume that all expressions are defined.

3a. x _ 15

· x 7 _ 2x

· 20 _ x 4

3b. 10x - 40 __ x 2 - 6x + 8

· x + 3 _ 5x + 15

2

322 Chapter 5 Rational and Radical Functions

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Example 3. Multiply. Assume that all expressions are defined

A. ! B. !

!!!!!!!!!!!Guided Practice. Multiply. Assume that all expressions are defined.

13. ! 14. !

!!!!!!!!! !!You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal. !!

3x5y3

2x3y7⋅10x

3y4

9x2y5x − 34x + 20

⋅ x + 5x2 − 9

x15

⋅ x7

2x⋅ 20x4

10x − 40x2 − 6x + 8

⋅ x + 35x +15

1 2

3 4 ÷ = 1

2 4 3

! 2

2 3 =

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Video Example 4. Divide. Assume that all expressions are defined.

A. ! B. !

!!!!!!!!!!!!!

!

57x2y

÷ 20x3

7xy4x5 − 9x3

x2 − x −12÷ x

5 − x4 −12x3

x2 −16

You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal.

4E X A M P L E Dividing Rational Expressions

Divide. Assume that all expressions are defined.

A 4 x 3 _ 9 x 2 y

÷ 16 _ 9 y 5

B x 5 - 4 x 3 _ x 2 - x - 2

÷ x 5 - x 4 - 2 x 3 __ x 2 - 1

4 x 3 _ 9 x 2 y

· 9 y 5

_ 16

Rewrite as

multiplication by the

reciprocal.

x 5 - 4 x 3 _ x 2 - x - 2

· x 2 - 1 __ x 5 - x 4 - 2 x 3

4 x 3 1 _ 9 x 2 y

· 9 y 5 4

_ 164

x 3 ( x 2 - 4)

_ x 2 - x - 2

· x 2 - 1 __ x 3 ( x 2 - x - 2)

x y 4

_ 4

x 3 (x - 2) (x + 2)

__ (x - 2) (x + 1)

· (x - 1) (x + 1)

__ x 3 (x - 2) (x + 1)

(x + 2) (x - 1)

__ (x + 1) (x - 2)

or x 2 + x - 2 _ x 2 - x - 2

Divide. Assume that all expressions are defined.

4a. x 2 _ 4

÷ x 4 y

_ 12 y 2

4b. 2 x 2 - 7x - 4 __ x 2 - 9

÷ 4 x 2 - 1 __ 8 x 2 - 28x + 12

5E X A M P L E Solving Simple Rational Equations

Solve. Check your solution.

A x 2 - 9 _ x + 3

= 7 B x 2 + 3x - 4 __ x - 1

= 5

(x - 3) (x + 3)

__ x + 3

= 7 Note thatx ≠ -3.

(x - 1) (x + 4)

__ x - 1

= 5 Note thatx ≠ 1.

x - 3 = 7 x + 4 = 5

x = 10 x = 1

Check −−−−−−−−−−

x 2 - 9 _ x + 3

= 7 Because the left side of the original equation is undefined when x = 1, there is no solution.

(10) 2 - 9 _

10 + 3 7

91 _ 13

7

7 7 ✔

Solve. Check your solution.

5a. x 2 + x - 12 __ x + 4

= -7 5b. 4 x 2 - 9 _ (2x + 3)

= 5

1 _ 2

÷ 3 _ 4

= 1 _ 2

· 4 _ 3

2 = 2 _

3

Check A graphing calculator shows that 1 is not a solution.

As you simplify a rational expression, take note of values that must be excluded. The excluded values are those that make the rational expression undefined.

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Example 4. Divide. Assume that all expressions are defined.

A. ! B. !

!!!!!!!!!!!!!Guided Practice. Divide. Assume that all expressions are defined.

15. ! 16. !

!!!!!!!!!!!!!!!

5x4

8x2y2÷ x4y12y2

2x2 − 7x − 4x2 − 9

÷ 4x2 −18x2 − 28x +12

x2

4÷ x4y12y2

2x2 − 7x − 4x2 − 9

÷ 4x2 −18x2 − 28x +12

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Video Example 5. Solve. Check your solution

A. ! B. !

!!!!!!!!!!!!!!!!!!!!!!

x2 − 25x − 5

= 8 x2 − 3x −10x + 2

= −7

Algebra II 5-2 Study Guide Page ! of !12 14

!

You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal.

4E X A M P L E Dividing Rational Expressions

Divide. Assume that all expressions are defined.

A 4 x 3 _ 9 x 2 y

÷ 16 _ 9 y 5

B x 5 - 4 x 3 _ x 2 - x - 2

÷ x 5 - x 4 - 2 x 3 __ x 2 - 1

4 x 3 _ 9 x 2 y

· 9 y 5

_ 16

Rewrite as

multiplication by the

reciprocal.

x 5 - 4 x 3 _ x 2 - x - 2

· x 2 - 1 __ x 5 - x 4 - 2 x 3

4 x 3 1 _ 9 x 2 y

· 9 y 5 4

_ 164

x 3 ( x 2 - 4)

_ x 2 - x - 2

· x 2 - 1 __ x 3 ( x 2 - x - 2)

x y 4

_ 4

x 3 (x - 2) (x + 2)

__ (x - 2) (x + 1)

· (x - 1) (x + 1)

__ x 3 (x - 2) (x + 1)

(x + 2) (x - 1)

__ (x + 1) (x - 2)

or x 2 + x - 2 _ x 2 - x - 2

Divide. Assume that all expressions are defined.

4a. x 2 _ 4

÷ x 4 y

_ 12 y 2

4b. 2 x 2 - 7x - 4 __ x 2 - 9

÷ 4 x 2 - 1 __ 8 x 2 - 28x + 12

5E X A M P L E Solving Simple Rational Equations

Solve. Check your solution.

A x 2 - 9 _ x + 3

= 7 B x 2 + 3x - 4 __ x - 1

= 5

(x - 3) (x + 3)

__ x + 3

= 7 Note thatx ≠ -3.

(x - 1) (x + 4)

__ x - 1

= 5 Note thatx ≠ 1.

x - 3 = 7 x + 4 = 5

x = 10 x = 1

Check −−−−−−−−−−

x 2 - 9 _ x + 3

= 7 Because the left side of the original equation is undefined when x = 1, there is no solution.

(10) 2 - 9 _

10 + 3 7

91 _ 13

7

7 7 ✔

Solve. Check your solution.

5a. x 2 + x - 12 __ x + 4

= -7 5b. 4 x 2 - 9 _ (2x + 3)

= 5

1 _ 2

÷ 3 _ 4

= 1 _ 2

· 4 _ 3

2 = 2 _

3

Check A graphing calculator shows that 1 is not a solution.

As you simplify a rational expression, take note of values that must be excluded. The excluded values are those that make the rational expression undefined.

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Example 5. Solve. Check your solution

A. ! B. !

!!!!!!!!!!!!!!!!!!!!!!!

x2 − 25x − 5

= 9 x2 + 3x −10x − 2

= 7

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Guided Practice. Solve. Check your solution

17. ! 18. !

!!!!!!!!!!!!!!!!!!!!5-2 Assignment (p 324) 18, 19, 24-26, 28-30, 32, 33, 36, 38, 40, 47-49.

x2 + x −12x + 4

= −7 4x2 − 92x + 3

= 5